CQ8

Q1

The first hint that you are given is that this is a critically damped spring. Therefore, what do we know about ζ ?

1

Q2

Now that we have our ζ , we can consider the next hint in the problem. We want our system to settle after 8 seconds ($T_{settle}$). For this, we want to use our time constant Tconstant equation, which states that

$T_{settle}=4T_{constant}=4ζω_n$

0.5

Q3

Now we have successfully obtained ζ and ωn, we can finish our calculations. Looking at the equation of the mass-spring-damper system and the ideal dynamics equations listed above, we can extract the following expressions by lining up the equivalent x parameters:

$\ddot x+\frac{c}{m}\dot x+\frac{k}{m}x=\frac{1}{m}f_{ext}$

$\ddot x+2\zeta\omega_n\dot x+\omega_n^2x=\omega_n^2x_d$

Therefore, we can see if these two equations are going to produce the same dynamic response, then for the term:

$\frac{c}{m}=2\zeta\omega_n$

and for the x term:

$\frac{k}{m}=2\zeta\omega_n$

Using these two equations (there are actually a few different equations that you can create, not just these two), we can reach our final solution. Solve for the spring constant k and the damping coefficient c.

k=4, c=16

Q4

Awesome!!! Great job!!! Now that we have $c$ and $k$, what does this mean? This means that if we create a critically damped mass-spring-damper system of mass 16, spring constant of 4, and damper coefficient of 16, the system will settle at $x_d$ after 8 seconds.

Keep this process in mind whenever you are working with systems like this. Now that you are familiar with this general approach, you should be able to solve lots of different versions of this question. For example, you could be asked to do the reverse and solve for the time constant, or you could be given parameters describing the system and be asked to solve for $x_d$, or you could be given these parameters, and asked to figure out what ζ is in order to classify the system (as overdamped, underdamped, or critically damped). You got this! 😃